Reversible Conservative Rational Abstract Geometrical Computation Is Turing-Universal

  • Jérôme Durand-Lose
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)


In Abstract geometrical computation for black hole computation (MCU ’04, LNCS 3354), the author provides a setting based on rational numbers, abstract geometrical computation, with super-Turing capability. In the present paper, we prove the Turing computing capability of reversible conservative abstract geometrical computation. Reversibility allows backtracking as well as saving energy; it corresponds here to the local reversibility of collisions. Conservativeness corresponds to the preservation of another energy measure ensuring that the number of signals remains bounded. We first consider 2-counter automata enhanced with a stack to keep track of the computation. Then we built a simulation by reversible conservative rational signal machines.


Abstract geometrical computation Conservativeness Rational numbers Reversibility Turing-computability 2-counter automata 


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  1. [Ada02]
    Adamatzky, A. (ed.): Collision based computing. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  2. [AM95]
    Asarin, E., Maler, O.: Achilles and the Tortoise climbing up the arithmetical hierarchy. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 471–483. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  3. [Ben73]
    Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 6, 525–532 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Ben88]
    Bennett, C.H.: Notes on the history of reversible computation. IBM J. Res. Dev. 32(1), 16–23 (1988)MathSciNetCrossRefGoogle Scholar
  5. [BNR91]
    Boccara, N., Nasser, J., Roger, M.: Particle-like structures and interactions in spatio-temporal patterns generated by one-dimensional deterministic cellular automaton rules. Phys. Rev. A 44(2), 866–875 (1991)CrossRefGoogle Scholar
  6. [Bou99]
    Bournez, O.: Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy. Theoret. Comp. Sci. 210(1), 21–71 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Čul87]
    Čulik II, K.: On invertible cellular automata. Complex Systems 1, 1035–1044 (1987)MathSciNetzbMATHGoogle Scholar
  8. [DL95]
    Durand-Lose, J.: Reversible cellular automaton able to simulate any other reversible one using partitioning automata. In: Baeza-Yates, R., Poblete, P.V., Goles, E. (eds.) LATIN 1995. LNCS, vol. 911, pp. 230–244. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  9. [DL97]
    Durand-Lose, J.: Intrinsic universality of a 1-dimensional reversible cellular automaton. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 439–450. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  10. [DL02]
    Durand-Lose, J.: Computing inside the billiard ball model. In: Adamatzky, A. (ed.) Collision-based computing, pp. 135–160. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. [DL05a]
    Durand-Lose, J.: Abstract geometrical computation 1: embedding black hole computations with rational numbers. Research Report RR-2005-05, LIFO, U. D’Orléans, France (2005),
  12. [DL05b]
    Durand-Lose, J.: Abstract geometrical computation for black hole computation. In: Margenstern, M. (ed.) MCU 2004. LNCS, vol. 3354, pp. 176–187. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. [DL05c]
    Durand-Lose, J.: Abstract geometrical computation: Turing-computing ability and undecidability. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 106–116. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. [DM02]
    Delorme, M., Mazoyer, J.: Signals on cellular automata. In: [Ada02], pp. 234–275 (2002)Google Scholar
  15. [Dub95]
    Dubacq, J.-C.: How to simulate Turing machines by invertible 1d cellular automata. International Journal of Foundations of Computer Science 6(4), 395–402 (1995)CrossRefzbMATHGoogle Scholar
  16. [FT82]
    Fredkin, E., Toffoli, T.: Conservative logic. International Journal of Theoretical Physics 21(3/4), 219–253 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Ila01]
    Ilachinski, A.: Cellular automata –a discrete universe. World Scientific, Singapore (2001)CrossRefzbMATHGoogle Scholar
  18. [JS90]
    Jacopini, G., Sontacchi, G.: Reversible parallel computation: an evolving space-model. Theoret. Comp. Sci. 73(1), 1–46 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [JSS02]
    Jakubowsky, M.H., Steiglitz, K., Squier, R.: Computing with solitons: a review and prospectus. In: [Ada 2002], pp. 277–297 (2002)Google Scholar
  20. [Kar90]
    Kari, J.: Reversibility of 2D cellular automata is undecidable. Phys. D 45, 379–385 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Kar94]
    Kari, J.: Reversibility and surjectivity problems of cellular automata. J. Comput. System Sci. 48(1), 149–182 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [Kar96]
    Kari, J.: Representation of reversible cellular automata with block permutations. Math. System Theory 29, 47–61 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Lec63]
    Lecerf, Y.: Machines de Turing réversibles. Récursive insolubilité en n ∈ N de l’équation u = θ n u, où θ est un isomorphisme de codes. Comptes rendus des séances de l’académie des sciences 257, 2597–2600 (1963)MathSciNetGoogle Scholar
  24. [LN90]
    Lindgren, K., Nordahl, M.G.: Universal computation in simple one-dimensional cellular automata. Complex Systems 4, 299–318 (1990)MathSciNetzbMATHGoogle Scholar
  25. [LTV98]
    Li, M., Tromp, J., Vitanyi, P.: Reversible simulation of irreversible computation. Physica. D 120, 168–176 (1998)CrossRefGoogle Scholar
  26. [Min67]
    Minsky, M.: Finite and infinite machines. Prentice Hall, Englewood Cliffs (1967)zbMATHGoogle Scholar
  27. [Mor92]
    Morita, K.: Computation-universality of one-dimensional one-way reversible cellular automata. Inform. Process. Lett. 42, 325–329 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Mor95]
    Morita, K.: Reversible simulation of one-dimensional irreversible cellular automata. Theoret. Comp. Sci. 148, 157–163 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [Mor96]
    Morita, K.: Universality of a reversible two-counter machine. Theoret. Comp. Sci. 168(2), 303–320 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [MT99]
    Mazoyer, J., Terrier, V.: Signals in one-dimensional cellular automata. Theoret. Comp. Sci. 217(1), 53–80 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [TM90]
    Toffoli, T., Margolus, N.: Invertible cellular automata: a review. Phys. D 45, 229–253 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [Tof77]
    Toffoli, T.: Computation and construction universality of reversible cellular automata. J. Comput. System Sci. 15, 213–231 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  33. [Tof80]
    Toffoli, T.: Reversible computing. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 632–644. Springer, Heidelberg (1980)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jérôme Durand-Lose
    • 1
  1. 1.Laboratoire d’Informatique Fondamentale d’OrléansUniversité d’OrléansORLÉANS

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